How do you know where to put the constant when finding general solutions for differential equations?

Let's suppose we want to solve with . Separating variables and integrating, we get



Leaving the solution in this form, the initial condition gives



This means the solution is .

Now if we were to write , then we would have found



so that the solution would have been .

But these two solutions are the same, since . So we get the same solution regardless of where we place , despite getting different values for .